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Persistence, Predictability, and Portfolio Planning

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Handbook of Quantitative Finance and Risk Management

Abstract

We use a model of stock price behavior in which the expected rate of return on stocks follows an Ornstein- Uhlenbeck process to show that levels of return predictability that cause large variation in valuation ratios and offer significant benefits to dynamic portfolio strategies are hard to detect or measure by standard regression techniques, and that the R 2 from standard short run predictive regressions carry little information about either long run predictability or the value of dynamic portfolio strategies. We propose a new approach to portfolio planning that uses forward-looking estimates of long run expected rates of return from dividend discount models. We show how such long run expected rates of return can be used to estimate the instantaneous expected rate of return under the assumption that the latter follows an Ornstein-Uhlenbeck process. Simulation results using four different estimates of long run rates of return on U.S. common stocks suggest that this approach may be valuable for long horizon investors.

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Notes

  1. 1.

    See, for example, Shiller (1981), LeRoy and Porter (1981), Kleidon (1986), Marsh and Merton (1986).

  2. 2.

    See, for example, Fama and French (1988b).

  3. 3.

    See, for example, Keim and Stambaugh (1986), Fama and French (1988a) and so forth.

  4. 4.

    See, for example, Lewellen (2004).

  5. 5.

    See Brav and Heaton (2002) and Lewellen and Shanken (2002).

  6. 6.

    Campbell (2001) shows that long-horizon regression tests have serious size distortions when asymptotic critical values are used, but some versions of such tests have power advantages remaining after size is corrected.

  7. 7.

    Campbell and Thompson (2004) show that the findings of Goyal and Welch (2004) are no longer true if sensible restrictions are imposed on the signs of predictive coefficients and return forecasts.

  8. 8.

    Campbell and Shiller (1998, 2001) and Fama and French (2002), among others, find that dividend yields have little or no forecast power for subsequent dividend growth in the years since 1870.

  9. 9.

    We shall find that this corresponds to the empirical estimates of this parameter reported below.

  10. 10.

    Boudoukh et al. (1993) report evidence that the ex ante risk premium is negative in some states of the world.

  11. 11.

    It is possible for innovations in discount rates to be positively correlated with current returns if the innovations in discount rates are highly correlated with innovations in cash flow news.

  12. 12.

    Of course, if the coefficient is not corrected for bias, the t-statistic is somewhat higher.

  13. 13.

    See Xia (2001) for an analysis of the effects of parameter uncertainty on optimal portfolio planning.

  14. 14.

    Note that we are assuming here that α = 0 and β = 1.

  15. 15.

    Note that P is used to denote the stock price with dividends reinvested and V to denote the ex-dividend value.

  16. 16.

    See Feller (1951). In our simulations below, we set \({\mu}^{{_\ast}} = -2.5\%\), which is about three standard deviations away from the long run \(\overline{\mu} = 9\%\).

  17. 17.

    The P/D ratio and dividend data are from Robert Shiller’s web page http://www.econ.yale.edu/shiller/data.htm. Fama and French (2002) report that the mean dividend growth rate for the S&P 500 was 2.74% for the period 1872 to 1950 and was 1.05% (1.3% from Shiller’s data) for the period 1951 to 2000, which contrasts with the real earnings growth rate of 2.82% over the last half century.

  18. 18.

    Brennan and Xia (2002) and Wachter (2002) show that the determination of an optimal consumption and investment program for an investor with time-additive utility can be represented as a time integral of optimal utility of terminal wealth problems.

  19. 19.

    For models of dynamic investment strategies with stochastic interest rates see Campbell and Viceira (2001) and Brennan and Xia (2002).

  20. 20.

    This was first derived by Kim and Omberg (1996); it is discussed in more detail in Xia (2001).

  21. 21.

    The out of sample RMSE is calculated from the simulated data used for Table 19.1; the predictive relation is estimated from the first 65 years of data, and the out-of-sample RMSE is calculated from the differences between the predicted and the realized returns over the following five years. The results, which are not reported here, are available on request. Goyal and Welch (2003, 2004) use the out-of-sample RMSE as a measure of the value of a predictive instrument.

  22. 22.

    Stambaugh (1999) and Barberis (2000) compare myopic and buy-and-hold strategies.

  23. 23.

    This is in contrast to the findings of Brennan and Torous (1999) who show that a buy-and-hold strategy performs well relative to a rebalancing strategy when the investment opportunity set is treated as constant.

  24. 24.

    Brennan and Xia (2001) assume a similar model for the dividend growth rate.

  25. 25.

    There is no assurance that κ t , the solution to Equation (19.22), will be equal to the expected rate of return except when all future dividends are known and discount rates are constant.

  26. 26.

    As noted above, 12.02% is the sample volatility of annual real dividend growth over the period 1872–2001, while 8.52% is the estimate of σ D ψ in Equation (19.25) derived from quarterly data for the period 1950.1 to 2002.2 using the algorithm described under Model 2 above.

  27. 27.

    The unconditional equity premium \(\overline{\gamma}\) for the unconditional strategy was calculated in three different ways: (1) using the same \(\overline{\mu}\) as that used in calculating the optimal strategy; (2) setting \(\overline{\gamma}\) to the sample mean of the S&P 500 Index excess return during the whole sample period of 1929 to 2002; and (3) setting \(\overline{\gamma}\) to the gradually updated sample mean excess return with the initial value calculated from 1929 to 1949 (or 1972 for the BGI or WA series). The first approach ensures that differences between the wealth realized under the optimal and the unconditional strategies are not caused by different assumptions about the unconditional equity premium. Unconditional strategies based on (2) and (3) have similar realized wealth as that based on (1) except for the period 1950–1970, during which unconditional strategies based on (2) and (3) significantly outperform that based on (1) but still underperform the optimal strategy. Since the relative performance of the optimal strategy is consistent across the three unconditional strategies, we only report results of the unconditional strategy based on (1) and omit those based on (2) and (3) for brevity.

  28. 28.

    The instantaneous expected returns estimated from the DDM long run expected returns depend on parameters of the expected return μ and the dividend growth g processes, κμ, σμ, and \(\overline{g}\) etc. These parameters were estimated using data from the whole sample period so that our estimates of instantaneous expected returns, even when they are based on real time DDM estimates, rely on future data. For the A&B and IL series, we also estimated μ for the period of 1950–2002 by first estimating the parameters using data only from 1900 to 1949, and the superior performance of the optimal strategy remains unchanged. We do not have long enough sample for the BGI and WA series, which are only available starting from 1973, to carry out this robustness check.

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Authors and Affiliations

  1. University of California, Los Angeles, CA, USA

    Michael J. Brennan

  2. Wharton School, Philadelphia, PA, USA

    Yihong Xia

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  1. Michael J. Brennan
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Editors and Affiliations

  1. Department of Finance and Economics, Rutgers University, 94 Rockafeller Road, Janice H. Levin Bldg., New Brunswick, NJ, 08854-8054, USA

    Cheng-Few Lee

  2. State Street Corp., Boston, MA, 94132, USA

    Alice C. Lee

  3. Center for PBBEF Research, North Brunswick, NJ, USA

    John Lee

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Brennan, M.J., Xia, Y. (2010). Persistence, Predictability, and Portfolio Planning. In: Lee, CF., Lee, A.C., Lee, J. (eds) Handbook of Quantitative Finance and Risk Management. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-77117-5_19

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